A Calabi theorem for solutions to the parabolic Monge–Ampère equation with periodic data

Abstract We classify all solutions to − u t det ⁡ D 2 u = f ( x )  in  R − n + 1 , where f ∈ C α ( R n ) is a positive periodic function in x . More precisely, if u is a parabolically convex solution to above equation, then u is the sum of a convex quadratic polynomial in x , a periodic function in x and a linear function of t . It can be viewed as a generalization of the work of Gutierrez and Huang in 1998. And along the line of approach in this paper, we can treat other parabolic Monge-Ampere equations.

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